Online presentations of finitely generated structures

Abstract We systematically investigate into the online content of finitely generated structures. The online content of a potentially infinite algebraic or combinatorial structure is perhaps best reflected by its PR-degrees (to be defined). We confirm a natural conjecture by showing that the PR-degrees of a finitely generated structure must be dense. Remarkably, we show that PR-degrees of an f.g. structure do not have to be upwards dense. As an application of our techniques, we refute a natural conjecture about honestly generated structures (to be stated).

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